Preheat the oven to 350ºF (180ºC). Line the bottom of an 8-inch square cake pan with parchment paper. Grease and flour the insides.
In a medium-sized bowl, whisk together the flour, cinnamon, baking soda, salt and nutmeg until well blended.
In a large bowl, beat the butter, with an electric mixer on medium speed, until pale and creamy, Add the brown sugar and oil. Continue beating until creamy and well blended, about 2-3 minutes. Beat in the eggs and vanilla extract in 3 separate additions, making sure each one is incorporated before adding the next. Stir in the mashed bananas. Add the flour mixture in two additions, followed by half of the sour cream after each addition. Mix just until combined.
Transfer the batter to the prepared cake pan. Bake until a toothpick inserted into the center of the cake comes out clean, about 25-30 minutes. Let the cake cool in the pan for 10 minutes, then remove it from the pan and place it on a cooling rack.